Software packages for the numerical solution of partial differential equations (PDEs) using the finite element method are important in different fields of research. The basic data structures and algorithms change in time, as the user\'s requirements are growing and the software must efficiently use the newest highly parallel computing systems. This is the central point of this work. To make efficiently use of parallel computing systems with growing number of independent basic computing units, i.e.~CPUs, we have to combine data structures and algorithms from different areas of mathematics and computer science. Two crucial parts are a distributed mesh and parallel solver for linear systems of equations. For both there exists multiple independent approaches. In this work we argue that it is necessary to combine both of them to allow for an efficient and scalable implementation of the finite element method. First, we present concepts, data structures and algorithms for distributed meshes, which allow for local refinement. The central point of our presentation is to provide arbitrary geometrical information of the mesh and its distribution to the linear solver. A large part of the overall computing time of the finite element method is spend by the linear solver. Thus, its parallelization is of major importance. Based on the presented concept for distributed meshes, we preset several different linear solver methods. Hereby we concentrate on general purpose linear solver, which makes only little assumptions about the systems to be solver. For this, a new FETI-DP (Finite Element Tearing and Interconnect - Dual Primal) method is proposed. Those the standard FETI-DP method is quasi optimal from a mathematical point of view, its not possible to implement it efficiently for a large number of processors (> 10,000). The main reason is a relatively small but globally distributed coarse mesh problem. To circumvent this problem, we propose a new multilevel FETI-DP method which hierarchically decompose the coarse grid problem. This leads to a more local communication pattern for solver the coarse grid problem and makes it possible to scale for a large number of processors. Besides the parallelization of the finite element method, we discuss an approach to speed up serial computations of existing finite element packages. In many computations the PDE to be solved consists of more than one variable. This is especially the case in multi-physics modeling. Observation show that in many of these computation the solution structure of the variables is different. But in the standard finite element method, only one mesh is used for the discretization of all variables. We present a multi-mesh finite element method, which allows to discretize a system of PDEs with two independently refined meshes. / Softwarepakete zur numerischen Lösung partieller Differentialgleichungen mit Hilfe der Finiten-Element-Methode sind in vielen Forschungsbereichen ein wichtiges Werkzeug. Die dahinter stehenden Datenstrukturen und Algorithmen unterliegen einer ständigen Neuentwicklung um den immer weiter steigenden Anforderungen der Nutzergemeinde gerecht zu werden und um neue, hochgradig parallel Rechnerarchitekturen effizient nutzen zu können. Dies ist auch der Kernpunkt dieser Arbeit. Um parallel Rechnerarchitekturen mit einer immer höher werdenden Anzahl an von einander unabhängigen Recheneinheiten, z.B.~Prozessoren, effizient Nutzen zu können, müssen Datenstrukturen und Algorithmen aus verschiedenen Teilgebieten der Mathematik und Informatik entwickelt und miteinander kombiniert werden. Im Kern sind dies zwei Bereiche: verteilte Gitter und parallele Löser für lineare Gleichungssysteme. Für jedes der beiden Teilgebiete existieren unabhängig voneinander zahlreiche Ansätze. In dieser Arbeit wird argumentiert, dass für hochskalierbare Anwendungen der Finiten-Elemente-Methode nur eine Kombination beider Teilgebiete und die Verknüpfung der darunter liegenden Datenstrukturen eine effiziente und skalierbare Implementierung ermöglicht. Zuerst stellen wir Konzepte vor, die parallele verteile Gitter mit entsprechenden Adaptionstrategien ermöglichen. Zentraler Punkt ist hier die Informationsaufbereitung für beliebige Löser linearer Gleichungssysteme. Beim Lösen partieller Differentialgleichung mit der Finiten Elemente Methode wird ein großer Teil der Rechenzeit für das Lösen der dabei anfallenden linearen Gleichungssysteme aufgebracht. Daher ist deren Parallelisierung von zentraler Bedeutung. Basierend auf dem vorgestelltem Konzept für verteilten Gitter, welches beliebige geometrische Informationen für die linearen Löser aufbereiten kann, präsentieren wir mehrere unterschiedliche Lösermethoden. Besonders Gewicht wird dabei auf allgemeine Löser gelegt, die möglichst wenig Annahmen über das zu lösende System machen. Hierfür wird die FETI-DP (Finite Element Tearing and Interconnect - Dual Primal) Methode weiterentwickelt. Obwohl die FETI-DP Methode vom mathematischen Standpunkt her als quasi-optimal bezüglich der parallelen Skalierbarkeit gilt, kann sie für große Anzahl an Prozessoren (> 10.000) nicht mehr effizient implementiert werden. Dies liegt hauptsächlich an einem verhältnismäßig kleinem aber global verteilten Grobgitterproblem. Wir stellen eine Multilevel FETI-DP Methode vor, die dieses Problem durch eine hierarchische Komposition des Grobgitterproblems löst. Dadurch wird die Kommunikation entlang des Grobgitterproblems lokalisiert und die Skalierbarkeit der FETI-DP Methode auch für große Anzahl an Prozessoren sichergestellt. Neben der Parallelisierung der Finiten-Elemente-Methode beschäftigen wir uns in dieser Arbeit mit der Ausnutzung von bestimmten Voraussetzung um auch die sequentielle Effizienz bestehender Implementierung der Finiten-Elemente-Methode zu steigern. In vielen Fällen müssen partielle Differentialgleichungen mit mehreren Variablen gelöst werden. Sehr häufig ist dabei zu beobachten, insbesondere bei der Modellierung mehrere miteinander gekoppelter physikalischer Phänomene, dass die Lösungsstruktur der unterschiedlichen Variablen entweder schwach oder vollständig voneinander entkoppelt ist. In den meisten Implementierungen wird dabei nur ein Gitter zur Diskretisierung aller Variablen des Systems genutzt. Wir stellen eine Finite-Elemente-Methode vor, bei der zwei unabhängig voneinander verfeinerte Gitter genutzt werden können um ein System partieller Differentialgleichungen zu lösen.

We present a numerical method to solve the optimal transport problem with a quadratic cost when the source and target measures are periodic probability densities. This method relies on a numerical resolution of the corresponding Monge-Ampère equation. We use an existing Newton-like algorithm that we generalize to the case of a non uniform final density. The main idea consists of designing an iterative scheme where the fully nonlinear equation is approximated by a non-constant coefficient linear elliptic PDE that we discretize and solve at each iteration, in two different ways: a second order finite difference scheme and a Fourier transform (FT) method. The FT method, made possible thanks to a preconditioning step based on the coefficient-averaged equation, results in an overall O(P LogP )-operations algorithm, where P is the number of discretization points. We prove that the generalized algorithm converges to the solution of the optimal transport problem, under suitable conditions on the initial and final densities. Numerical experiments demonstrating the robustness and efficiency of the method on several examples of image processing, including an application to multiple sclerosis disease detection, are shown. We also demonstrate by numerical tests that the method is competitive against some other methods available.

Finite-difference numerical methods are developed for the solution of some systems in the biomedical sciences; namely, a predator-prey model and the SEIR (Susceptible/Exposed/ Infectious/Recovered) measles model. First-order methods are developed to solve the predator-prey model and one second-order method is developed to solve the SEIR measles model. The predator-prey model is extended to one-space dimension to incorporate diffusion. The SEIR measles model is extended to one-space dimension to incorporate (i) diffusion, (ii) convection and (iii) diffusion-convection. The SEIR measles model is extended further to model diffusion in two-space dimensions. The reaction terms in these systems of partial differntial equations contain nonlinear expressions. Nevetheless, it is seen that the numerical solutions are obtained by solving a linear algebraic system at each time step, as opposed to solving a nonlinear algebraic systems, which is often required when integrating non-linear partial differential equations. The development of each numerical method is made in the light of experience gained in solving the system of ordinary differential equations for each system. The numerical methods proposed for the solution of the initial-value problem for the predator-prey and measles models are characterized to be implicit. However, in each case it is seen that the numerical solutions are obtained explicitly. In a series of numerical experiments, in which the ordinary differential equations are solved first of all, it is seen that the proposed methods have superior stability properties to those of the well-known, first-order, Euler method to which they are compared. Incorporating the proposed methods into the numerical solution of partial differential equations is seen to lead to economical and reliable methods for solving the systems.

The dynamics of the world's oceans occur at a vast range of length scales. Although there are theories that aid in understanding the dynamics at planetary scales and microscales, the motions in between are still not yet well understood. This work discusses a numerical model to study barotropic wind-driven gyre flow that is capable of resolving dynamics at the synoptic, O(1000 km), mesoscale, O(100 km) and submesoscales O(10 km). The Quasi-Geostrophic (QG) model has been used predominantly to study ocean circulations but it is limited as it can only describe motions at synoptic scales and mesoscales. The Rotating Shallow Water (SW) model that can describe dynamics at a wider range of horizontal length scales and can better describe motions at the submesoscales. Numerical methods that are capable of high-resolution simulations are discussed for both QG and SW models and the numerical results are compared. To achieve high accuracy and resolve an optimal range of length scales, spectral methods are applied to solve the governing equations and a third-order Adams-Bashforth method is used for the temporal discretization. Several simulations of both models are computed by varying the strength of dissipation. The simulations either tend to a laminar steady state, or a turbulent flow with dynamics occurring at a wide range of length and time scales. The laminar results show similar behaviours in both models, thus QG and SW tend to agree when describing slow, large-scale flows. The turbulent simulations begin to differ as QG breaks down when faster and smaller scale motions occur. Essential differences in the underlying assumptions between the QG and SW models are highlighted using the results from the numerical simulations.

Currently, in the field of reactor physics, there is a drive for high fidelity, numerical simulations of reactors for the purposes of design and analysis. Since the behavior of a reactor is dependent on various physical phenomena, high fidelity simulations must be able to accurately couple these different types of physics. This is the essence of multiphysics simulations. In order to accurately simulate the thermal behavior of a reactor, the physics of neutron transport must be coupled to the fluid flow and solid phase conduction occurring within the reactor. This thesis develops a computational fluid dynamics solver for this purpose. The solver is based on the PCICE solution algorithm and employs cell-centered finite volumes. In addition to the fluid dynamics solver, a newly developed form of conjugate heat transfer is implemented. This implementation tightly couples the physics of solid phase heat conduction with the fluid dynamics in an efficient and consistent manner. Finally, the radiation transport code EVENT is used to provide heat generation data to the fluids solver. Using this fluids solver, several benchmark problems are analyzed and the formulation is validated.

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[pt] O tamanho das zonas plásticas (zps) presentes na ponta de trincas valida a utilização da Mecânica da Fratura Linear Elástica (MFLE). Dessa forma, a partir das estimativas dessas zps, este trabalho estuda o limite de validade dos dois parâmetros que caracterizam MFLE. Esses dois parâmetros são o Fator de Intensidade de Tensões (K) e a T-stress. Este trabalho mostra que esses dois parâmetros são termos da expansão da série de Williams partir da função de tensão de Westergaard. As duas formas são maneiras diferentes de se obter a solução linear elástica (LE) completa para o campo de tensões gerados na ponta de trincas. Esta tese mostra que esses dois campos de tensões têm uso limitado, pois eles geram tensões infinitas na ponta de trincas. Essas tensões singulares são características do problema matemático, não reproduzindo o real comportamento mecânico. Devido a isso, o problema das estimativas das zps é intrinsecamente não linear. Como tentativa de contornar o problema, este trabalho propõe três maneiras de considerar os efeitos do escoamento nas estimativas zps em que se adota um material perfeitamente plástico. As estimativas feitas por campos LE são verificadas numericamente a partir do uso do Método dos Elementos Finitos (MEF) e do Método Híbrido dos Elementos de Contorno (MHEC). Duas das propostas de considerar os efeitos do escoamento nas zps são utilizadas juntamente com MHEC. Como contribuição final, este trabalho estima zps a partir de uma análise numérica não linear via MEF em que os efeitos do encruamento também são testados. Essas estimativas são comparadas com as estimativas LE corrigidas em que se considera um material perfeitamente plástico. / [en] The size of plastic zones (pz) at the crack tips validates the use of Linear Elastic Fracture Mechanics (LEFM). Thus, this thesis studies the limits of validity of the two parameters that characterize MFLE from the pz estimates. These two parameters are the stress intensity factor (K) and the T-stress. This work shows that KI and the T-stress are terms of the Williams’ series expansion, which is the complete linear elastic (LE) solution for the stress fields generated at the crack tips. It also demonstrates that the Williams’ series is a different way of writing the Westergaard stress function in terms of a trigonometric series with infinite terms, and comments that even if the two functions are the complete LE solution for cracked bodies, they have limited use, because they generate infinite tensions at the crack tip. These singular stresses are characteristics of mathematical problem, not reproducing the real mechanical behavior. As an attempt to outline the problem of singularity in a qualitative way, this work proposes three ways to consider the yielding effects in pz estimates in which one adopts a perfectly plastic material. The completeness of the stress fields generated by the Westergaard stress function is verified numerically from the use of Finite Element Method (FEM) and from of the Hybrid Boundary Element Method (HBEM). Two of the proposals to consider the yielding effects in the pz are used in conjunction with HBEM. The problem of pz estimates is instrinsically non-linear due to the singularity obtained by the mathematical formulation. Finally, this thesis also estimates the pz from a non-linear numerical analysis via FEM. The hardening effects are also tested in these nonlinear estimates. Moreover, they are compared to estimates corrected LE in which a perfectly plastic material is considered.

Taludes escavados em filitos muitas vezes estão sujeitos à instabilidade, principalmente quando há outras famílias de descontinuidades presentes na estrutura. Este problema está presente na mina de Candiota, onde foi identificado, em um dos taludes, potencialidade para ruptura por tombamento. Este trabalho aborda uma discussão sobre métodos analíticos e numéricos aplicado a tombamento primário, com objetivo de avaliar o fator de segurança do talude em estudo por meio de método numérico por elementos finitos, utilizando o Phase2 da Rocscience, e métodos analíticos. Também são apresentadas doze simulações para avaliar se o movimento de tombamento em modelagens numéricas obedece a um efeito de escala. Com uso de métodos numéricos dois modelos foram construídos: um contínuo equivalente e outro contínuo (mas com as descontinuidades inseridas no modelo). No modelo com as descontinuidades a rocha foi tratada como elástica por meio do critério de Hoek-Brown e as descontinuidades foram tratadas como elasto-plásticas por meio do critério de Mohr-Coulomb. Os resultados das modelagens da mina de Candiota foram semelhantes e mostraram fatores de segurança em níveis de estabilidade, sendo que no modelo contínuo (mas com descontinuidades inseridas) se observou de forma mais clara as tensões cisalhantes induzidas nas descontinuidades quando ocorre o movimento de tombamento. O uso dos métodos analíticos mostrou-se ineficientes para o problema apresentado. Os resultados das doze modelagens sugerem que o aumento do espaçamento das descontinuidades influencia nos fatores de resistência dos taludes e na forma do movimento de tombamento. / Slopes in phyllites are often subject to instability, especially when there are other families of discontinuities in the structure. This problem is present in Candiota mine, which was identified in one of the slopes potential to toppling. This work show a discussion of analytical and numerical methods applied to toppling, to evaluate the slope safety factor being studied by numerical method finite element using the Phase2 of Rocscience, and analytical methods. Also shown are twelve simulations to evaluate if the toppling in numerical modeling follows a scaling effect. With use of numerical methods two models were built: an equivalent continuous and another solid (but with discontinuities inserted in the model). In the model with the discontinuities the rock was treated as elastic by the Hoek-Brown criterion and discontinuities were treated as elastic-plastic by the Mohr-Coulomb criterion. The results of Candiota mine were similar and showed safety factors in stability levels, and in the continuous model (with discontinuities) was observed more clearly the shear stress induced in discontinuities when the movement of overturning occurs. The use of analytical methods proved inefficient for the problem presented. The results of the twelve modeling suggests increasing the spacing of the discontinuities of slope influences the resistance factor and the form of the toppling movement.

[pt] Neste trabalho é apresentado o estudo da interação entre um jato supersônico e uma superfície plana, com o objetivo de analisar o comportamento do campo de velocidade, pressão e temperatura do escoamento. Este estudo encontra sua motivação no processo de descamação térmica de rochas duras, a qual pode resultar da iteração entre um jato a alta pressão e temperatura e a rocha. Este processo, que pode ser útil na perfuração de rochas duras e profundas, ocorre devido ao acúmulo de tensões térmicas na rocha, o qual pode acarretar sua fratura. Este tipo de processo também envolve diversos mecanismos aerodinâmicos e termodinâmicos, que são isoladamente fenômenos abertos. No desenvolvimento deste trabalho o escoamento foi modelado pelas equações de Navier - Stokes bidimensionais para uma mistura de gases perfeitos em um sistema de coordenadas cilíndrico. O modelo considerado para descrever o transporte turbulento é o modelo de uma equação de Spalart - Allmaras, o qual envolve a solução de uma equação diferencial para a viscosidade turbulenta. Estas equações são resolvidas utilizando-se uma metodologia de volumes finitos adaptada a escoamentos compressíveis. A descrição dos escoamentos transientes obtidos necessitou de diversas modificações ao código computacional existente. Estas modificações trataram, em particular, das condições de contorno, que utilizam a noção de características, e do modelo de turbulência. A estrutura do escoamento resultante da interação entre o jato supersônico e a parede é estudada, avaliando-se a influência (i) da distância entre a saída do jato e a parede, (ii) da razão de pressões entre o jato e o ambiente. Além disso, é examinada a evolução transiente do escoamento. Os resultados obtidos são analisados com vista a obter as melhores condições aerodinâmicas para o processo de descamação térmica. / [en] I in this work a study of the interaction between a supersonic jet and a planar surface is presented, with the aim to analyze the behavior of the velocity, pressure and temperature flowfield. This study finds its motivation in the process of thermal spallation of hard rocks, which may result from the interaction between a high pressure and high temperature jet and the rock. This process, that can be used in the drilling of hard and deep rocks, occurs due to the accumulation of thermal stresses in the rock, which can cause its fracture. This type of process also involves several aerodynamic and thermodynamic mechanisms, which are still open phenomena. In the development of this work the flow was modeled by the two-dimensional Navier-Stokes equation for a mixture of perfect gases in a cylindrical coordinates system. The model considered to describe the turbulent transport is the one equation of Spalart - Allmaras model, which involves the solution of a differential equation for the turbulent viscosity. These equations are solved using a finite volumes methodology which is adapted to compressible flows. The description of the obtained transient flow required several modifications in the existing computational code. These modifications involved, in particular, the choice of boundary conditions, that use the notion of characteristics, and the turbulence model. The structure of the flow resulting from the interaction between the supersonic jet and the wall is studied. In particular, are examined the influence (i) the distance between the jet and wall, (II) of the pressures ratio between the jet and the environment. Moreover, the transient evolution of the flow is examined. The obtained results are examined to determine the best aerodynamic conditions for the process of thermal spallation to occur.

La résolution d’équations différentielles de divers degrés de complexité est nécessaire afin de simuler tous les phénomènes présents dans les écoulements complexes de turbomachine et en particulier les effets hors équilibre qui peuvent y jouer un rôle prépondérant. Aujourd’hui, seule l’approche LES (Large Eddy Simulation) sous forme totalement compressible permet d’obtenir avec une précision satisfaisante la physique associée aux écoulements réactifs et turbulents en géométrie complexe. Le travail porte sur la modélisation numérique et physique des échanges thermiques en proche paroi. Ce travail de thèse s'est appuyé sur le projet Européen COPA-GT dédié à la simulation numérique et multi-physique d'un moteur complet. / The resolution of differential equations of diverse degree of complexity is necessary to simulate the phenomena present in the complex turbomachinery flows and in particular, requires accounting for unsteady effects that may have a preponderant role. Today, only the LES (Large Eddy Simulation) fully compressible approach has the required accuracy to predict the physics associated to reactive and turbulent flows in such complex geometries. This work covers the numerical modelling of physics in the near-wall region of a high-pressure turbine blade with special focus on thermal predictions. This work was supported by the European project COPA-GT, dedicated to the numerical multi-physics simulation of a complete gas turbine.